In Exercises 83–94, find each product. [5y + (2x+3)][5y − (2x+3)]

Verified step by step guidance

1. Recognize the expression as a difference of squares: $(a+b)(a-b)=a^2-b^2$.
2. Identify $a$ and $b$ in the expression: $a=5y$ and $b=(2x+3)$.
3. Apply the difference of squares formula: $(5y{)}^2-(2x+3{)}^2$.
4. Calculate $(5y{)}^2$ which is $25y^2$.
5. Calculate $(2x+3{)}^2$ using the formula $(m+n{)}^2=m^2+2mn+n^2$.

Key Concepts

Factoring Difference of Squares

The expression given is in the form of a difference of squares, which follows the formula a^2 - b^2 = (a + b)(a - b). In this case, the terms 5y and (2x + 3) can be identified as 'a' and 'b', respectively. Recognizing this pattern allows for simplification and efficient calculation of the product.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property is essential for expanding expressions where terms are multiplied across parentheses. In the context of the given expression, applying the distributive property will help in multiplying the two binomials to find the resulting polynomial.

Combining Like Terms

After expanding the expression, combining like terms is necessary to simplify the result. Like terms are terms that have the same variable raised to the same power. This process ensures that the final expression is in its simplest form, making it easier to interpret and use in further calculations.